This fact has already been observed [1]} and in this paper, we aim at providing more details on this subject. The point of view that we take, where we consider a covariant 4D extension of the fracton transformation of a rank-2 symmetric tensor (given in
(REF )), has not been explored in the literature.
We want to study the resulting tensor gauge field theory, using standard Quantum Field Theory techniques. The starting point is the necessary gauge fixing procedure, which already displays some non-standard features, on which we will elaborate. We will then discuss in which sense fractons and gravitons are related, what commonalities they have, and how they differ. We will then consider the case where the fractons are massive, also exhibiting some unexpected features.
The first notable difference with LG is that we have at our disposal a dimensionless parameter \(a\) , and second that in order to find the propagators, the gauge fixing is much simpler. Nevertheless the propagator structure turns out to be quite rich and its study leads to a “phase” diagram of the theory which includes particular values of the free parameter \(a\)  where the invariant action acquires a different structure.
In our analysis we will adopt the simplest possible Landau gauge which has the advantage of making the “physics” more transparent and the computations much easier. Particular attention has ben paid to the degrees of freedom (dof) of the theory as the \(a\) -parameter moves on the real line. We shall also see that the phase diagram suggests an interpretation of the \(a\) -parameter and a possible connection with fractons. At the end we will explore, as is usual in LG, the massive extension of the model and will find that, in order to avoid tachyonic poles, a segment of the real line must be excluded from the domain of \(a\) . The paper is organised as follows. In Section 2 the invariant action for a 4D symmetric tensor field is derived as the most general one invariant under the particular infinitesimal symmetry characterising the “emergent electromagnetism” described in [2]}, [1]}. We then analyze the gauge fixing procedure, and we show that the Landau gauge choice is the natural one for this model. Finally, we compute the propagators, which lead to identify some critical values of the parameter \(a\) . One of these special values, notably \(a=0\) , is expected since the model acquires a stronger symmetry and hence a completely different structure and while another one is also expected where the symmetric tensor becomes traceless, there is a third intriguing singular point, \(a=1\) , where the model can be written in terms of a non propagating, invariant vector field. In Section 3 the degrees of freedom are studied, which allow for a further interpretation of the \(a\) -parameter, which turns out to be related also to the gauge fixing structure of the theory, \(i.e.\)  the choice of a scalar ghost. The massive case is considered in Section 4, where appear analogies and differences with respect to LG: as in the Fierz-Pauli theory of LG only one mass parameter is allowed instead of two [4]}, [5]}, as a symmetric tensor field would in principle admit, and, differently to what happens in massive LG [6]}, [7]}, [8]}, the number of degrees of freedom does not depend on the presence of a mass term. Finally, in Section 5 we discuss the main feature of fractons, \(i.e.\)  the fact of having limited or even zero mobility [9]}, [10]}, [11]}, and we propose a parameter, which of course is a function of \(a\) , which measures the rate of propagation of the particle described by this theory. We show that indeed for a particular value of this parameter the corresponding particle does not propagate, which lead us to identify the parameter, hence the theory, corresponding to fractons.
